Constructive analysis

In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics. This contrasts with classical analysis, which (in this context) simply means analysis done according to the (more common) principles of classical mathematics.

Generally speaking, constructive analysis can reproduce theorems of classical analysis, but only in application to separable spaces; also, some theorems may need to be approached by approximations. Furthermore, many classical theorems can be stated in ways that are logically equivalent according to classical logic, but not all of these forms will be valid in constructive analysis, which uses intuitionistic logic.

For a simple example, consider the intermediate value theorem (IVT). In classical analysis, IVT says that, given any continuous function f from a closed interval [a,b] to the real line R, if f(a) is negative while f(b) is positive, then there exists a real number c in the interval such that f(c) is exactly zero. In constructive analysis, this does not hold, because the constructive interpretation of existential quantification ("there exists") requires one to be able to construct the real number c (in the sense that it can be approximated to any desired precision by a rational number). But if f hovers near zero during a stretch along its domain, then this cannot necessarily be done.

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